%I #32 Nov 14 2022 00:36:37

%S 1,1,1,1,7,1,1,28,28,1,1,84,336,84,1,1,210,2520,2520,210,1,1,462,

%T 13860,41580,13860,462,1,1,924,60984,457380,457380,60984,924,1,1,1716,

%U 226512,3737448,9343620,3737448,226512,1716,1,1,3003,736164,24293412,133613766,133613766,24293412,736164,3003,1

%N Triangle T(n,m) read by rows: T(n,m) = Product_{i=0..5} binomial(n+i,m)/binomial(m+i,m).

%C Triangle of generalized binomial coefficients (n,k)_6; cf. A342889. - _N. J. A. Sloane_, Apr 03 2021

%C The matrix inverse starts

%C 1;

%C -1, 1;

%C 6, -7, 1

%C -141, 168, -28, 1;

%C 9911, -11844, 2016, -84, 1;

%C -1740901, 2081310, -355320, 15120, -210, 1. - _R. J. Mathar_, Mar 22 2013

%H Seiichi Manyama, <a href="/A142465/b142465.txt">Rows n = 0..139, flattened</a>

%H Johann Cigler, <a href="https://arxiv.org/abs/2103.01652">Pascal triangle, Hoggatt matrices, and analogous constructions</a>, arXiv:2103.01652 [math.CO], 2021.

%H Johann Cigler, <a href="https://www.researchgate.net/publication/349376205_Some_observations_about_Hoggatt_triangles">Some observations about Hoggatt triangles</a>, Universität Wien (Austria, 2021).

%F T(n,m) = A056941(n,m)*binomial(n+5,m)/binomial(m+5,m).

%F Sum_{k=0..n} T(n, k) = A005364(n).

%e Triangle begins as:

%e 1;

%e 1, 1;

%e 1, 7, 1;

%e 1, 28, 28, 1;

%e 1, 84, 336, 84, 1;

%e 1, 210, 2520, 2520, 210, 1;

%e 1, 462, 13860, 41580, 13860, 462, 1;

%e 1, 924, 60984, 457380, 457380, 60984, 924, 1;

%e 1, 1716, 226512, 3737448, 9343620, 3737448, 226512, 1716, 1;

%e 1, 3003, 736164, 24293412, 133613766, 133613766, 24293412, 736164, 3003, 1;

%p A142465 := proc(n,m)

%p mul(binomial(n+i,m)/binomial(m+i,m),i=0..5) ;

%p end proc; # _R. J. Mathar_, Mar 22 2013

%t T[n_, k_]:= Product[Binomial[n+j, k]/Binomial[k+j, k], {j,0,5}];

%t Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten

%o (PARI) T(n, k) = prod(j=0, 5, binomial(n+j, k)/binomial(k+j, k)); \\ _Seiichi Manyama_, Apr 01 2021

%o (Magma)

%o A142465:= func< n,k | (&*[Binomial(n+j,k)/Binomial(k+j,k): j in [0..5]]) >;

%o [A142465(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Nov 13 2022

%o (SageMath)

%o def A142465(n,k): return product(binomial(n+j,k)/binomial(k+j,k) for j in (0..5))

%o flatten([[A142465(n,k) for k in range(n+1)] for n in range(13)]) # _G. C. Greubel_, Nov 13 2022

%Y Cf. A001263, A005364 (row sums), A056941, A056940, A056939, A142467.

%Y Triangles of generalized binomial coefficients (n,k)_m (or generalized Pascal triangles) for m = 1,...,12: A007318 (Pascal), A001263, A056939, A056940, A056941, A142465, A142467, A142468, A174109, A342889, A342890, A342891.

%K nonn,easy,tabl

%O 0,5

%A _Roger L. Bagula_, Sep 20 2008, Jan 28 2009

%E Edited by the Associate Editors of the OEIS, May 17 2009